Computation of Eigenvalues and Eigenvectors of a Mistuned Bladed Disk Via Unidirectional Taylor Series Expansions

2010 ◽  
Vol 132 (4) ◽  
Author(s):  
Alok Sinha
Keyword(s):  
Taylor Series ◽  
Taylor Series Expansion ◽  
Series Expansions ◽  
Eigenvalues And Eigenvectors ◽  
Numerical Examples ◽  
Bladed Disk ◽  
Repeated Eigenvalues ◽  
Taylor Series Expansions ◽  
Derivatives Of ◽  
Computation Of Eigenvalues

This paper deals with the computation of eigenvalues and eigenvectors of a mistuned bladed disk. First, the existence of derivatives of repeated eigenvalues and corresponding eigenvectors is discussed. Next, an algorithm is developed to compute these derivatives. It is shown how a Taylor series expansion can be used to efficiently compute eigenvalues and eigenvectors of a mistuned system. Numerical examples are presented to corroborate the validity of theoretical analysis.

Computation of Eigenvalues and Eigenvectors of a Mistuned Bladed Disk

2006 ◽  
Author(s):  
Alok Sinha
Keyword(s):  
Series Expansion ◽  
Taylor Series ◽  
Taylor Series Expansion ◽  
Eigenvalues And Eigenvectors ◽  
Numerical Examples ◽  
Bladed Disk ◽  
Repeated Eigenvalues ◽  
Derivatives Of ◽  
Computation Of Eigenvalues

This paper deals with fundamental aspects of variations in eigenvalues and eigenvectors of a bladed disk due to mistuning. First, the existence of derivatives of repeated eigenvalues and corresponding eigenvectors is thoroughly examined. Next, an algorithm is developed to compute these derivatives. It is shown how a Taylor series expansion can be used to efficiently compute eigenvalues and eigenvectors of a mistuned system. This methodology is developed for perturbations in both repeated and unrepeated eigenvalues of the tuned system. Lastly, numerical examples are presented.

The Derivatives of Repeated Eigenvalues and Their Associated Eigenvectors

1996 ◽  
Vol 118 (3) ◽  
pp. 390-397 ◽  
Author(s):  
M. I. Friswell
Keyword(s):  
Taylor Series ◽  
Design Parameters ◽  
Eigenvalues And Eigenvectors ◽  
First Order ◽  
Repeated Eigenvalues ◽  
Eigenvector Derivatives ◽  
Eigenvalue And Eigenvector ◽  
Derivatives Of

This paper considers the calculation of eigenvalue and eigenvector derivatives when the eigenvalues are repeated. An extension to Nelson’s method is used to calculate the first order derivatives of eigenvectors when the derivatives of the associated eigenvalues are also equal. The continuity of the eigenvalues and eigenvectors is discussed, and the discontinuities in the eigenvectors, when they are regarded as functions of two or more design parameters, is demonstrated. The validity of Taylor series for the eigenvalues and eigenvectors is examined and the use of these series critically assessed.

scholarly journals Approximate pricing of European and Barrier claims in a local-stochastic volatility setting

2017 ◽  
Vol 04 (02n03) ◽  
pp. 1750018
Author(s):  
Weston Barger ◽  
Matthew Lorig
Keyword(s):  
Stochastic Volatility ◽  
Taylor Series ◽  
Asymptotic Expansions ◽  
Diffusion Coefficients ◽  
Series Expansions ◽  
Underlying Asset ◽  
Numerical Examples ◽  
Correlation Parameter ◽  
Taylor Series Expansions

We derive asymptotic expansions for the prices of a variety of European and barrier-style claims in a general local-stochastic volatility setting. Our method combines Taylor series expansions of the diffusion coefficients with an expansion in the correlation parameter between the underlying asset and volatility process. Rigorous accuracy results are provided for European-style claims. For barrier-style claims, we include several numerical examples to illustrate the accuracy and versatility of our approximations.

scholarly journals A Shortcut to LAD Estimator Asymptotics

1991 ◽  
Vol 7 (4) ◽  
pp. 450-463 ◽  
Author(s):  
P.C.B. Phillips
Keyword(s):  
Asymptotic Expansion ◽  
Regression Model ◽  
Taylor Series ◽  
Asymptotic Theory ◽  
Random Variables ◽  
Generalized Functions ◽  
Infinite Variance ◽  
Series Expansions ◽  
Linear Functionals ◽  
Taylor Series Expansions

Using generalized functions of random variables and generalized Taylor series expansions, we provide quick demonstrations of the asymptotic theory for the LAD estimator in a regression model setting. The approach is justified by the smoothing that is delivered in the limit by the asymptotics, whereby the generalized functions are forced to appear as linear functionals wherein they become real valued. Models with fixed and random regressors, and autoregressions with infinite variance errors are studied. Some new analytic results are obtained including an asymptotic expansion of the distribution of the LAD estimator.

Full-Scale Bounds Estimation for the Nonlinear Transient Heat Transfer Problems with Interval Uncertainties

2021 ◽  
pp. 2150026
Author(s):  
Ruifei Peng ◽  
Haitian Yang ◽  
Yanni Xue
Keyword(s):  
Heat Transfer ◽  
Series Expansion ◽  
Taylor Series ◽  
Taylor Series Expansion ◽  
Transient Heat Transfer ◽  
High Fidelity ◽  
Interval Scale ◽  
Nonlinear Transient ◽  
Transfer Problems ◽  
Derivatives Of

A package solution is presented for the full-scale bounds estimation of temperature in the nonlinear transient heat transfer problems with small or large uncertainties. When the interval scale is relatively small, an efficient Taylor series expansion-based bounds estimation of temperature is stressed on the acquirement of first and second-order derivatives of temperature with high fidelity. When the interval scale is relatively large, an optimization-based approach in conjunction with a dimension-adaptive sparse grid (DSG) surrogate is developed for the bounds estimation of temperature, and the heavy computational burden of repeated deterministic solutions of nonlinear transient heat transfer problems can be efficiently alleviated by the DSG surrogate. A temporally piecewise adaptive algorithm with high fidelity is employed to gain the deterministic solution of temperature, and is further developed for recursive adaptive computing of the first and second-order derivatives of temperature. Therefore, the implementation of Taylor series expansion and the construction of DSG surrogate are underpinned by a reliable numerical platform. The parallelization is utilized for the construction of DSG surrogate for further acceleration. The accuracy and efficiency of the proposed approaches are demonstrated by two numerical examples.

scholarly journals A Method for the Generation of Various Ghost Correction Algorithms—the Example of the Positivity Method and the Exponential Method

1991 ◽  
Vol 13 (4) ◽  
pp. 199-212 ◽  
Author(s):  
P. Van Houtte
Keyword(s):  
Pole Figure ◽  
Taylor Series ◽  
The Other ◽  
New Method ◽  
Series Expansions ◽  
Theoretical Scheme ◽  
Taylor Series Expansions ◽  
Exponential Method ◽  
Pole Figure Data

A theoretical strategy is presented that can derive the algorithms of several existing ghost correction methods. The examples of the positivity method and the “GHOST” method are elaborated. A new method is derived as well: the “exponential” method. It can successfully replace the quadratic method as a method that yields an exactly non-negative complete C.O.D.F. from pole figure data. The theoretical scheme that can generate all these algorithms makes use of the fact, that several parameter sets can be defined in order to describe a C.O.D.F. The parameters of one set are then functions of those of the other. The algorithms are derived from Taylor series expansions of these functions.

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